Have a look at www.trenitalia.com/en/index.html for your train schedules.
Alitalia flies directly from Trieste to Milan Malpensa twice daily, but I wouldn't call it them "cheap flights" - around 106 euros. The flight only takes about an hour though, so it's up to you to weigh up the cost v time factor after looking at the Trenitalia website!
(And further to your other post, Trieste is great - well, we think so. We always try to get there for at least one night when we're in northern Italy. I don't really know why we like the place so much - probably something to do with all of those wonderful historic cafes - but we just do! So I hope your brother and his family - and you - enjoy it too...)
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